An affine connection is symplectic if \(\nabla _{i}\Omega _{jk} = 0\) . For background about symplectic connections see [1]} or [2]} and the references therein. The space \(\mathbb {T}(M, \Omega )\)  of symplectic affine connections on \((M, \Omega )\)  is an affine space modeled on the vector space \(\Gamma (T^{\ast }M\otimes S^{2}(T^{\ast }M))\)  of covariant tensors \(\Pi _{ijk} = \Pi _{i(jk)}\)  symmetric in the last two entries. Precisely, if \(\nabla \in \mathbb {T}(M, \Omega )\)  and \(\bar{\nabla }= \nabla + \Pi _{ij}\,^{k}\)  then \(\bar{\nabla }_{i}\Omega _{jk} = -2\Pi _{i[jk]}\) , so that \(\bar{\nabla }\in \mathbb {T}(M, \Omega )\)  if and only if \(\Pi _{i[jk]} = 0\) . Note that \(\Pi _{ip}\,^{p} =0\) , reflecting that a symplectic affine connection preserves the associated volume form \(\Omega _{n}\) . The affine subspace \(\mathbb {S}(M, \Omega )\subset \mathbb {T}(M, \Omega )\)  comprising torsion-free symplectic affine connections is modeled on the vector space \(\mathsf {S}^{3}(M)\) , for the difference tensor of torsion-free connections satisfies \(\Pi _{[ij]}\,^{k} = 0\) , and with \(\Pi _{i[jk]} = 0\) , this implies \(\Pi _{ijk} = \Pi _{(ijk)}\) . The space \(\mathbb {S}(M, \Omega )\)  is nonempty, for if \(\bar{\nabla }\)  is any torsion-free affine connection then \(\nabla = \bar{\nabla }+ \tfrac{2}{3}\Omega ^{kp}\bar{\nabla }_{(i}\Omega _{j)p}\)  is symplectic. It is convenient to write \(T_{\nabla }\mathbb {T}(M, \Omega ) = \Gamma (T^{\ast }M\otimes S^{2}(T^{\ast }M))\)  or \(T_{\nabla }\mathbb {S}(M, \Omega )= \mathsf {S}^{3}(M)\)  for the tangent space to the affine spaces \(\mathbb {T}(M, \Omega )\)  or \(\mathbb {S}(M, \Omega )\)  at \(\nabla \) . The focus here is on \(\mathbb {S}(M, \Omega )\) , and it is mainly in Section REF  that reference will be made to \(\mathbb {T}(M, \Omega )\) . Throughout this paper symplectic connection means a torsion-free symplectic connection, while symplectic affine connection means a connection preserving \(\Omega \)  but possibly having torsion.
